Grassmannian G_{2,4} and S^2 \times S^2

Hi,

I am currently working on a physics problem involving a Grassmannian,
namely the real Grassmannian G_{2,4}. My work in so far has been on the
Plücker embedding of the Grassmannian in RP^5, but I'm looking for a
simpler formulation.

I've heard this Grassmannian is basically S^2 times S^2: how do I see
that? Is there a way of putting charts on the manifold that makes this
structure transparent; perhaps even so transparent that I can induce a
metric tensor from (R^4 times R^4, delta) which would look like the two
2-spheres?

I am thankful for any answers!

//Mehring


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